How to make a shortcut key for switching to Texture paint mode?
There are no exact hotkeys for mode change. But you can assign custom keys by left-clicking on mode then click add shortcut and press key on keyboard to assign.

I recommend, before adding key as shortcut, go to user preferences->input-> change filter to keybinding then press shortcut to see what are assigned to it.
 

There is more related info in following link:
https://blenderartists.org/t/how-to-set-custom-shortcuts-for-mode-switching/626237